Rewriting Creation and Annihilation OperatorsFor context, I am studying the quantum Hall Effect, integer and fractional, and this came up in my instructors online notes. I am unsure why you would rewrite this, and further what is the best way to approach it. To expand on my idea, I was thinking to expand $\pi$ to $\vec{\pi}=m\vec{v}=\vec{p}-\frac{q}{c}\vec{A}$ and perhaps try from there.

Projectile motion, canon vs cliffWorking on this now. Also, the velocity of the ball at the top of the cliff is zero because if it were not, it would go higher than 85m; the question tells you it just reaches the top.

Additional mass of block on inclined planeThanks for showing your work Justin. In your original work I believe you did not cancel a factor of $g$ and did not evaluate the sine properly. The cloth is a ruse to get you to ignore friction effects. Friction problems of this caliber will usually tell you how to obtain friction or just give you a coefficient of friction.